This is the problem: Given the ODE $x'=f(x)$, suppose that $f:\mathbb R^2\rightarrow\mathbb R^2$, and $f$ satisfies the hipoteses of Poincaré-Bendixson Theorem, if exists $V:\mathbb R^2\rightarrow\mathbb R$ is constant over the solutions, i.e., $V$ is a first integral, then $f$ has no cicle limit.

I think that I didn't understand the question, becouse by hipoteses, the $w$-limit is compact, not empty, and don't have critical points, so by the theorem the only possible $\omega$-limits are periodic orbits, so how I should prove that is not a periodic orbit to?

Thanks in advance.

  • $\begingroup$ What do you mean by $V$ is constant over the solutions? Is $V$ the potential function? $\endgroup$ – Alex R. Jun 10 '16 at 21:46
  • $\begingroup$ See these notes: users.ecs.soton.ac.uk/mb8/lect4.pdf Closed orbits are impossible in a gradient system. $\endgroup$ – Alex R. Jun 10 '16 at 21:49
  • $\begingroup$ $V$ is not necessary the potential function, $V$ is a first integral for the system. $\endgroup$ – e.turatti Jun 11 '16 at 16:04

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